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1.5 Elementary Matrices and

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1 1.5 Elementary Matrices and
-1 a Method for Finding A

2 Definition An n×n matrix is called an elementary matrix if it can be obtained from the n×n identity matrix by performing a single elementary row operation.

3 Example1 Elementary Matrices and Row Operations

4 Theorem 1.5.1 Row Operations by Matrix Multiplication
If the elementary matrix E results from performing a certain row operation on and if A is an m×n matrix ,then the product EA is the matrix that results when this same row operation is performed on A . When a matrix A is multiplied on the left by an elementary matrix E ,the effect is to performan elementary row operation on A .

5 Example2 Using Elementary Matrices

6 Inverse Operations If an elementary row operation is applied to an identity matrix I to produce an elementary matrix E ,then there is a second row operation that, when applied to E, produces I back again. Table 1.The operations on the right side of this table are called the inverse operations of the corresponding operations on the left.

7 Example3 Row Operations and Inverse Row Operation
The 2 ×2 identity matrix to obtain an elementary matrix E ,then E is restored to the identity matrix by applying the inverse row operation.

8 Theorem 1.5.2 Every elementary matrix is invertible ,and the inverse is also an elementary matrix.

9 Theorem 1.5.3 Equivalent Statements
If A is an n×n matrix ,then the following statements are equivalent ,that is ,all true or all false.

10 Row Equivalence Matrices that can be obtained from one another by a finite sequence of elementary row operations are said to be row equivalent . With this terminology it follows from parts (a )and (c ) of Theorem that an n×n matrix A is invertible if and only if it is row equivalent to the n×n identity matrix.

11 A method for Inverting Matrices

12 Example4 Using Row Operations to Find (1/3)
Find the inverse of Solution: To accomplish this we shall adjoin the identity matrix to the right side of A ,thereby producing a matrix of the form we shall apply row operations to this matrix until the left side is reduced to I ;these operations will convert the right side to ,so that the final matrix will have the form

13 Example4 Using Row Operations to Find (2/3)

14 Example5 Showing That a Matrix Is Not Invertible

15 Example4 Using Row Operations to Find (3/3)

16 Example6 A Consequence of Invertibility

17 Exercise Set 1.5 Question 3

18 Exercise Set 1.5 Question 10

19 Exercise Set 1.5 Question 17

20 Exercise Set 1.5 Question 22

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